Why exactly do radioactive elements decay in half every set amount of time? What is happening on an atomic scale?
In: Physics
It is the mathematical result if each atom has some probability of decaying each unit of time.
If each atom has a 1% each of decaying each second that result in a calculable half-life. The atoms have no memory so each second they exit the probability is the same
1% chance of decay means that 99% survived
Start with 10 000 atoms and after 1 s you 10 000*0.99 = 9900 left. After 2 seconds you have 9900*0.99= 9801 atoms left.
But 9900*0.99 = 10 000*0.99*0.99 = 10 000* 0.99^2
if you expand it you get to after n seconds you have 10 000* 0.99^n atoms left.
If you solve the equation 10 000* 0.99^n= 5000 and if you put it into for example wolfram alpha you get n=68.97 less call it 69 seconds.
The result is if 1% of all atoms decay each second the half-life is 69 seconds.
Half- life is a simple thing to use in the calculation you can for example direct know that after two half-life you have 1/4 of the original amount so 2500 after 138 s.
You can even calculate the amount that you have after 20 s The formula is:
initial amount*(1/2)^( time/ half-life)
So you get 10000*(1/2)^(20/69) = 8179 atoms
Probabilities like this work if you have a lot of atoms and there is thousand of billion of a billion atom if a gram of matter.
The do not work well if you have very few. You cant tell the time it takes for a single atom to decay but you can say that if you have 1 billion you only have half a billion after a half-life.
Atoms decay because of the quantum effect in the interaction of the week and strong force in the nucleus of an atom. The result of it that each atom has a percentage chance of decaying in a unit of time.
You can do the exact same thing for an increase. If you have 100 in a bank account and 6% interest each year you can calculate the [Doubling_time](https://en.wikipedia.org/wiki/Doubling_time) that will be 11.9 years
You can calculate the amount you have after any amount if time with the formula:
initial amount*(2)^( time/ doubling time)
So if the doubling time is 11.9 year s and you start with 100 you have after 1 year 100*2^(1/11.9)=105.9977 = 106 So a 6% annual interest. IT is not exact because 11.9 year is founded to 2 decimals places it is more exactly 11.8956610459…..
So you get a half-life if you lose a percentage after a unit of time and a doubling time if it increases by a percentage after a unit of time.
The effect of band account interest and radioactive decay are mathematically almost identical
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